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Let M(x8) be "x has sent y an e-mail message" and T(x,y) be "x has telephoned y,"
where the universe of discourse consists of all students in your class. Use
quantifiers to express the following statements. (Assume that all e-mail messages
that were sent are received.)
(a) (3%) There is a student in your class who has sent everyone else in your class
an e-mail message.
(b) (3%) Every student in the class has either received an e-mail message or
received a telephone call from another student in the class.
2. (5%) How many nonisomorphic simple graphs are there with 4 vertices? Justify
your answer.
3. A rooted tree is called an m-ary tree if every internal vertex has no more than rn
children. A rooted m-ary tree is called afull m-aly tree if every internal vertex has
exactly m children. A rooted m-ary tree of height h is balanced if all leaves are at
levels h or h-1. Consider a full m-ary tree T that has 8 1 leaves and height 4.
(a) (5%) Give the upper and lower bounds for m.
(b) (3%) What is m if T is also balanced?
4. Letf be a function from the set A to the set B. Let S be a subset of B. We define the
inverse image of S to be the subset of A containing all pre-images of all elements
of S. We denote the inverse image of S by f -'(Q, so that f -'(Q = {a E A (a) E
S ) . Let g(x) = M.Find
If
(a) (2%) g -'({OH.
(b) (2%) g -'({-I, 0,l)).
(c) (2%) -l({x I 0 < x < 1)).
5. (7%) Prove that the sum of the cubes of three consecutive positive integers is
divisible by 9.
6. (a) (4%) Assume that there are 100 students of different heights, fkom which two
groups of 10 students each are selected. In how many ways can the selection
be made so that the tallest student in the first group is shorter than the shortest
student in the second group?
(b) (4%) Use a combinatorial argument to show that C(2n, 2)=2C(ny2)+n2.
7. (6%) Suppose we choose 101 nu.mbers (without repetition) fiom the integers 1 to
200. Show that, among the chosen numbers, there exist two such that one divides
another.
8. (4%) Let a, be the number of subsets of the set (1, 2, ..., r ) that do not contain
two consecutive numbers. Derive a recurrence relation for a,.
9. (a) (4%) Explain what "a sorting method is stable" means.
(b) (4%) Is Heap Sort a stable sorting method? If yes, please prove your answer. If
no, please give an example to justify your answer.
10. (13%) Show that the number of distinct binary trees with n internal nodes is
1 1. (4%) A linear list is maintained circularly in an array q [ ~1,.
,
.,n - 11 to represent
a circular queue. The variable front is used to point to one position from the
first element in the circular queue. Another variable rear is used to point to the
last element in the circular queue. Obtain a formula in terms of front, rear and
n for the number of elements in the circular queue.
12. (4%) Give, using 'big oh' notation, the tightest worst case time complexity of the
following procedures as a function of n.
function recursive (n : integer): integer;
begin
if n <= 1 then
return (2)
else
return( 2 * recursive( n /2) + 2 * recursive( n 12 ))
end
13. (4%) Draw the Max-heap after the following operations are performed beginning
with an empty heap: insert 34, insert 39, insert 24, delete-max, insert 5, insert 35,
insert 17, delete-max.
'
'
14. (4%) Let a hash table and a hashing Eunction be defmed as follows:
var table : array [0..7] of hashnode;
hashing function: h(x)= mid-squaring of x;
Mids-squaring is defined as:
1. squaring x
2. using the second rightmost digit (in octal) of the result in step 1 to obtain the
bucket address.
Let the sequence inserted to the hash table be: insert 23, insert 14, insert 8, insert
18, insert 33. With linear probing as rehash technique, draw the closed hash
table,
15. (13%) We have a directed graph G = ( Y E) represented using adjacency lists. The
edge costs are integers in the range {1,2,3,4, 5). Assume that G has no self-loops
or multiple edges. Design an algorithm that solves the single-source shortest path
problem on G in o(IvI + 1~1).